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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 9}\\
%{\Large DRAFT/not a final version}

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%\item In addition to the residual $U(1)_{em}$ gauge symmetry
%associated with the electromagnetic interaction, the standard model
%has some global symmetries (phase transitions related to lepton and
%baryon number) surviving the spontaneous breaking of $SU(2)_L \times
%U(1)_Y$. Find them, and interpret them. Is it possible to break them?

%\item (a) How is the SM modified if one adds a $SU(2)_L$ singlet
%neutrino?  Derive all additional terms. (b) Show that you can make the
%neutrino massive by not adding any particles, but just by giving a
%Majorana mass term for the left-handed neutrino.

\item 
%As in lectures (sec. 7.2.1), derive the formula for differential
% cross section. 
Show that
\begin{displaymath}
(2 \pi) ^4 \delta ^4 (p_3+p_4-p) \frac{d^3 \vec p_3}{(2 \pi)^3 2 E_3}
\frac{d^3\vec p_4 }{(2 \pi)^3 2 E_4 }
\end{displaymath}
is Lorentz invariant. The cross section is generally given by
\begin{displaymath}
d \sigma=\frac {(2\pi)^4 \delta^4 (p_1+p_2-p_f) }{\sqrt {(p_1 \cdot
p_2)^2-m_1^2 m_2^2}} \left| \langle f | T | i \rangle \right| ^2 \prod_f
{\frac{ d^3 \vec p _f}{(2 \pi)^3 2 E_j}} .
\end{displaymath}
Use this to derive the cross section for scattering $a_1+a_2
\rightarrow b_1+b_2$ in the center of mass frame:
\begin{displaymath}
\frac {d \sigma}{d \Omega}= \frac {\left| \langle f | T | i \rangle \right| ^2}{64 \pi ^2 s}
\left( \frac{|\vec p _f |}{ | \vec p _i |} \right) ,
\end{displaymath}
where $d \Omega$ is the element of solid angle about some outgoing
particle (see e.g. Halzen-Martin exercise 4.2, or Bjorken-Drell,
Cheng-Li app. A, Perle chapter 6).  Express the initial and final
momenta by using the Mandelstam variables $s$, $t$, $u$.  \\ Hint:
Show $d^3p/(2E)=\delta(p^2-m^2) \Theta(p_0) d^4 p$ (Renton
p. 152). See Halzen-Martin section 4.3.

%\item (solving this problem will give extra points) Derive carefully
%the differential cross section of the Compton scattering of
%unpolarized photon and electron (section 7.2.2 of Chaichian and
%Nelipa). Show that at non-relativistic limit ($E_\gamma \rightarrow
%0$) one obtains the classical Thomson formula
%\begin{displaymath}
%\sigma=\frac {8 \pi} 3 \frac{\alpha^2}{m_e^2} .
%\end{displaymath}

\item (solving this problem will give extra points) Draw the
lowest-order Feynman diagrams for the pair annihilation process in QED
\begin{displaymath}
e^+(p_1,s_1)+e^-(p_2,s_2) \rightarrow \gamma(k_1,\varepsilon_1) +
\gamma(k_2,\varepsilon_2) .
\end{displaymath}
Write down the transition amplitude using Feynman rules. Calculate the
differential cross section of the process as a function of Mandelstam
variables $s$, $t$, $u$ (taking the electron mass into account). You
can assume that the initial electrons are unpolarized. Verify that at
high-energy limit the squared amplitude is proportional to
\begin{displaymath}
\frac ut + \frac tu .
\end{displaymath}
What would the angular distribution of outgoing photons look like in
high-energy $e^+ e^-$ collider, like LEP? \\ Hint: See the lectures,
section 7.2.2, and exercise 6.22 of Halzen and Martin.

\item Derive the total cross section for the neutral current ($Z$
exchange) scattering $\nu_\mu +e^- \rightarrow \nu_\mu +e^-$ in the
laboratory frame of reference (=initial electron at rest). \\ Hint:
Renton p. 277, Okun p. 130, 201, Cheng-Li p. 365. You may assume that
$q^2<<M_Z^2$ and ignore mass of the electron in the {\em final} state
(why?).


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Return the solved problems to the downstairs box in Laurikainen
building on Monday November 16th at 12 o'clock at latest. There are
two problems solving sessions: Tuesdays 16-18 in the Exercise room
(harjoitussali) in the basement of the Laurikainen bldg [English] and
Fridays 10-12 in Hall 5, Tallqvist bldg [Finnish]. \\ The www-page of
this course is: http://www.physics.helsinki.fi/{\%}7Epuolamak/hitu/

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